Solveeit Logo
Login

Question

Mathematics Question on Some Properties of Definite Integrals

The value of (1+sin⁡2π9+icos⁡2π91+sin⁡2π9−icos⁡2π9)3\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3 is

A

−12(1−i3)-\frac{1}{2}(1-i \sqrt{3})

B

12(3+i)\frac{1}{2}(\sqrt{3}+i)

C

−12(3−i)-\frac{1}{2}(\sqrt{3}-i)

D

12(1−i3)\frac{1}{2}(1-i \sqrt{3})

Answer

−12(3−i)-\frac{1}{2}(\sqrt{3}-i)

Explanation

Solution

The correct answer is (C) : −12(3−i)-\frac{1}{2}(\sqrt{3}-i)
Let sin92π​+icos92π​=z
(1+zˉ1+z​)3=(1+z1​1+z​)3=z3
⇒(i(cos92π​−isin92π​))3
=−i(cos32π​−isin32π​)=−i(2−1​−i2√ 3​​)
⇒2−1​(√ 3​−i).